Conclusion

Conclusion

The construction connects a finite Wilson probability law to a common continuum hierarchy through explicit kernels, nonlinear integrations, calibrated histories and physical coefficient spaces. The full mesh limit at each suitable fixed period precedes the outer volume refinement. Reflection positivity, the phase algebra and the tested boundary construction then give the fields on their common cyclic domain. The operator-product and contact constructions retain their finite horizons and chosen devices on that same hierarchy.

The stress calculation returns to actual finite local functions, controls their equation residuals and fixes the centered trace amplitude. Its logarithmic short-distance asymptotic yields the finite mass-shell and Maxwell comparison conclusions with the hypotheses stated above. No positive mass gap is asserted.

This paper provides the full natural-language mathematical exposition of the expanded source. The flagship in the same Zenodo release [2] develops Rodgers’s source–readout realization and recovery of this construction. Both papers retain the definitions and derivations required for their own arguments.